Is running cloth physics without animation, or an endless non-looping animation possible?
Unfortunately, this is not possible. The simulation is geared towards having a 'start' point and moving on frame by frame, storing the result in the Cloth Cache so that it can be replayed at a later point in time - there is no facility for a 'continuous' simulation. Even if there was, you would have the difficulty of stopping the simulation at the desired point in time and capturing the 'state' - currently this is achieved by simply scrubbing back through the frames until you get to the desired point and then 'applying' the cloth modifier to capture that frame from the cache and apply it to the mesh (or as a Shape Key).

So, yes, your best way forward is to increase the length of the simulation to give you enough time to tinker with the pinned vertices to get to the desired state before pausing the simulation to capture your 'frame'.

However, that said, once your cloth simulation has been baked you can use the Particle Edit mode to manipulate the mesh manually - this could be a way of achieving your goal. ie, run the simulation to get it 'almost there' and then use the Particle Edit to 'comb' the cloth (and the paths each vertex takes) into shape (as you would with Hair).